Without loss of generality we can assume \(A_3 > 0\) . The change of variable
\(\tau = A_{3}^{\frac{\alpha _1+\alpha _2}{m}}t^{\alpha _1+\alpha _2}\)  allows us to express the integral in (REF ) as
\(\frac{1}{(\alpha _1+\alpha _2)A_3^{\frac{\alpha _1+\alpha _2}{m}}} \int _{0}^{A_{3}^{\frac{\alpha _1+\alpha _2}{m}}} e^{-i(a \tau ^{\gamma _1} + b \tau ^{\gamma _2} + \tau ^{\gamma _3})} \, d\tau ,\) 
with \(a:=A_1 A_3^{-\frac{\alpha _1}{m}}\) , \(b:= A_2 A_3^{-\frac{\alpha _1}{m}}\) , \(\gamma _j := \frac{\alpha _j}{\alpha _1+\alpha _2}\)  for \(j=1,2\) 
and \(\gamma _3:= \frac{m}{\alpha _1+\alpha _2}\) . Let \(\Phi (\tau ) = a \tau ^{\gamma _1} + b \tau ^{\gamma _2} + \tau ^{\gamma _3}\) . To prove the
lemma it is enough to see that there exists a positive constant \(C\)  independent of \(a\) , \(b\) , and \(A_3\)  such that for all \(B > 1\)
\( \left| \int _{1}^{B} e^{-i \Phi (\tau )} \, d\tau \right| \le C.\) 
For \(\gamma \in \mathbb {R}\)  and \(j \in \mathbb {N}\) , we set \(\gamma (j) = \gamma (\gamma - 1) \cdot \cdot \cdot (\gamma - j +1)\) . We observe that \(\tau ^{j} \Phi ^{(j)}(\tau ) = \gamma _{1}(j) a \tau ^{\gamma _1} + \gamma _{2}(j) b \tau ^{\gamma _2} + \gamma _{3}(j) \tau ^{\gamma _3}\) ,
\(j=1,2,3\) , and that the matrix equation
\(\begin{bmatrix} \gamma _1(1) & \gamma _1(2) & \gamma _1(3) \\ \gamma _2(1) & \gamma _2(2) & \gamma _2(3) \\ \gamma _3(1) & \gamma _3(2) &\gamma _3(3) \end{bmatrix}\begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\) 
has unique solution \((c_1, c_2, c_3)\)  (since \(\det \left[ \gamma _i(j) \right] = \gamma _1 \gamma _2 \gamma _3(\gamma _2 - \gamma _1)(\gamma _3 - \gamma _1) (\gamma _3 - \gamma _2) \ne 0\) ). Thus \(\tau ^{\gamma _3} = \sum _{j=1}^{3} c_j \tau ^{j} \Phi ^{(j)}(\tau )\)  and so
\(\tau ^{\gamma _3} \le \sum _{j=1}^{3} |c_j \tau ^{j} \Phi ^{(j)}(\tau )|\)  for all \(\tau \ge 1\) . Then, \((1, B) \subset \cup _{j=1}^{3} I_j\) 
where \(I_j = \lbrace  \tau \in (1,B) : |\Phi ^{(j)}(\tau )| \ge 1/3|c_j| \rbrace \) . Since \(\frac{d}{d\tau } \left( \tau ^{\gamma _1-\gamma _2+1}\frac{d}{d\tau } \left( \tau ^{-\gamma _1 + j +1} \frac{d}{d\tau } \Phi ^{(j)}(\tau ) \right) \right) \ne 0\)  for all \(\tau \ge 1\)  and each
\(j=1,2,3\) , the Rolle's Theorem applied twice implies that, for each \(c \in \mathbb {R}\) , the equation \(\Phi ^{(j)}(\tau ) =c\)  has at most
three solutions. Hence, each \(I_j\)  has at most six connected components. Being \(m \ge 3(\alpha _1+\alpha _2)\)  we have \(\gamma _3 \ge 3\) ,
so (REF ) follows from the Van der Corput lemma (see e.g. [1]}, p. 332).

